Floor function

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Research consensus and handling negative numbers

$\newcommand{\Floor}[1]{ {\text{Floor}{\left({#1}\right)} } }$

Definition

For $x\in\mathbb{R}_{\ge 0}$ there is no variation on the meaning of the floor function, however for negative numbers there are varying conventions.

Non-negative

Defined as follows:

• $\text{Floor}:\mathbb{R}_{\ge 0}\rightarrow\mathbb{N}_0$ by $\text{Floor}:x\mapsto$$\text{Max}$$(T_x)$ where $T_x:\eq\big\{n\in\mathbb{N}_0\ \big\vert\ n\le x\big\}\subseteq\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0}$ - note that the maximum element is defined as $T_x$ is always finite.
• This has the property that $x\le\Floor{x}$.

Negative numbers

Researching this opened my eyes to a massive dispute.... consensus seems to be that $x\le \Floor{x}$ is maintained, rounding is a separate and massive issue!